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A333849

a(n) = gcd(A333848(n), 2*(2*n+1)), for n >= 0.

2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 6, 1, 2, 1, 2, 1, 2, 2, 2, 6, 2, 1, 2, 1, 1, 2, 2, 10, 6, 1, 2, 6, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 6, 2, 2, 2, 2, 1, 6, 1, 2, 6, 2, 2, 6, 2, 1, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 2, 2, 6, 2, 1, 2, 10, 2, 2, 2, 1, 10, 1, 2, 18, 2, 2, 2, 1, 2

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OFFSET

0,1

COMMENTS

For n >= 1, a(n) enters the formula for the length L(2*n+1) = A332441(n) of the directed Euler tour ET(2*n+1, q0 = 1) based on the unsigned Schick sequence for 2*n+1, namely L(2*n+1) = A003558(n)*2*(2*n+1)/a(n). For Schick sequences and references see A332439.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.

FORMULA

a(n) = gcd(A333848(n), 2*(2*n+1)), for n >= 0.

MATHEMATICA

{2}~Join~Table[GCD[Total@ Select[Range[1, m, 2], GCD[#, m] == 1 &], 2 m], {m, Array[2 # + 1 &, 85]}] (* Michael De Vlieger, Oct 15 2020 *)

PROG

(PARI) f(n) = if (n==0, 0, my(m=2*n+1); vecsum(select(x->((gcd(m, x)==1) && (x%2)), [1..m]))); \\ A333848

a(n) = gcd(f(n), 2*(2*n+1)); \\ Michel Marcus, May 05 2020

CROSSREFS

Cf. A003558, A332439, A332441, A333848.